Document Type
Conference Proceeding
Publication Date
4-25-2017
Abstract
This paper investigates connections between algebraic structures that are common in theoretical computer science and algebraic logic. Idempotent semirings are the basis of Kleene algebras, relation algebras, residuated lattices and bunched implication algebras. Extending a result of Chajda and Länger, we show that involutive residuated lattices are determined by a pair of dually isomorphic idempotent semirings on the same set, and this result also applies to relation algebras. Generalized bunched implication algebras (GBI-algebras for short) are residuated lattices expanded with a Heyting implication. We construct bounded cyclic involutive GBI-algebras from so-called weakening relations, and prove that the class of weakening relation algebras is not finitely axiomatizable. These algebras play a role similar to representable relation algebras, and we identify a finitely-based variety of cyclic involutive GBI-algebras that includes all weakening relation algebras. We also show that algebras of down-closed sets of partially-ordered groupoids are bounded cyclic involutive GBI-algebras.
Recommended Citation
Jipsen P. (2017) Relation Algebras, Idempotent Semirings and Generalized Bunched Implication Algebras. In: Höfner P., Pous D., Struth G. (eds) Relational and Algebraic Methods in Computer Science. RAMICS 2017. Lecture Notes in Computer Science, vol 10226. Springer, Cham. doi: 10.1007/978-3-319-57418-9_9
Peer Reviewed
1
Copyright
Springer
Comments
This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Lecture Notes in Computer Science, volume 10226, in 2017 following peer review. The final publication is available at Springer via DOI:10.1007/978-3-319-57418-9_9.